I believe so. For example,
A similar arguments can be used for graphs (as in graph theory), non-binary trees, heaps, stacks, linked lists etc.
Indeed, before programming languages offered advanced data structures, sophisticated objects and types, recursion (and much more) -- all the data -- had to be stored in (organized) arrays. Trees, hash tables etc. were simulated by using arrays and pointers. Even recursion.
Are there exceptions? In some ways, we are getting back to the old times, with unstructured data, such as member postings on social networks. Although structuring unstructured data (by putting it into clusters and taxonomies) allow it to be manipulated much more easily.
The answer is yes since 1936. Any program with any data can be represented as binary array. First Turing and then Post published their article in that year. We call these things Turing and Post-Turing machines.